A237597 a(n) = |{0 < k < prime(n): n divides pi(k*n)}|, where pi(.) is given by A000720.

1, 1, 1, 2, 2, 1, 1, 3, 4, 3, 3, 3, 2, 4, 3, 3, 5, 7, 1, 3, 3, 5, 2, 5, 4, 4, 5, 5, 3, 7, 3, 2, 3, 4, 8, 4, 2, 6, 4, 5, 6, 8, 7, 2, 8, 2, 7, 1, 3, 6, 4, 6, 5, 1, 7, 4, 4, 3, 5, 6, 4, 8, 6, 5, 2, 5, 8, 4, 2, 5, 7, 5, 3, 1, 3, 2, 6, 3, 2, 4

Offset: 1

Author

Zhi-Wei Sun, Feb 10 2014

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for all n > 0.

See also A237614 for the least k > 0 with pi(k*n) divisible by n.

Examples

			a(6) = 1 since pi(11*6) = 3*6 with 11 < prime(6) = 13.
a(19) = 1 since pi(33*19) = 6*19 with 33 < prime(19) = 67.
a(759) = 1 since pi(2559*759) = 191*759 with 2559 < prime(759) = 5783.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..2500

Crossrefs

Cf. A000040, A000720, A237578, A237598, A237612, A237614.

Programs

Mathematica

a[n_]:=Sum[If[Mod[PrimePi[k*n],n]==0,1,0],{k,1,Prime[n]-1}]
Table[a[n],{n,1,80}]